Central Limit Theorem Statistics Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

hard
A uniform distribution has ฮผ=5\mu = 5 and ฯƒ=2.89\sigma = 2.89. For samples of size 36, what are the mean and standard error of the sampling distribution? Is it approximately normal?

Solution

  1. 1
    Step 1: Mean of sampling distribution = ฮผ=5\mu = 5. SE = 2.8936=2.896โ‰ˆ0.48\frac{2.89}{\sqrt{36}} = \frac{2.89}{6} \approx 0.48.
  2. 2
    Step 2: Since n=36โ‰ฅ30n = 36 \geq 30, the CLT applies: the sampling distribution is approximately normal.

Answer

Mean = 5, SE โ‰ˆ 0.48, approximately normal by CLT.
Even though the population is uniform (flat, not bell-shaped), the CLT ensures that sample means from samples of 36 are approximately normally distributed.

About Central Limit Theorem

The Central Limit Theorem (CLT) states that for sufficiently large sample sizes (usually nโ‰ฅ30n \geq 30), the sampling distribution of the sample mean xห‰\bar{x} is approximately normal, regardless of the shape of the original population distribution.

Learn more about Central Limit Theorem โ†’

More Central Limit Theorem Examples

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